A well-known cooling structure for an internal combustion engine includes a radiator-routing passage by way of a radiator and a bypass passage that bypasses the radiator. The radiator-routing passage and the bypass passage form a coolant circulation path through which coolant is circulated by a water pump through water jackets in a cylinder portion and a cylinder head portion of the engine. The cooling structure further includes a thermostat valve that changes over between the circulation through the radiator-routing passage and the circulation through the bypass passage (see, for example, Patent Document 1).